The Hornich-Hlawka Inequality and Bernstein Functions

The Hornich-Hlawka inequality for three real numbers is extended from the identity function to all Bernstein functions on the half-line. For vectors in a Euclidean space it is shown to hold for the square-root function

Stable tail dependence functions : some basic properties

We prove some important properties of the extremal coefficients of a stable tail dependence function (“STDF”) and characterise logistic and some related STDFs. The well known sufficient conditions for composebility of logistic STDFs are shown to be also necessary

Higher order monotonic (multi-) sequences and their extreme points

Functions on the half-line which are non-negative and decreasing of a higher order have a long tradition. When normalized they form a simplex whose extreme points are well-known. For functions on N0 = {0, 1, 2, . . .} the situation is different. Since an n-monotone sequence is in general not the restriction of an n-monotone function on R+ (apart from n = 1 and n = 2), it is not even clear at the beginning if the normalized n-monotone sequences form a simplex. We will show in this paper that this is actually true, and we determine their extreme points. A corresponding result will also be proved for multi-sequences. The main ingredient in the proof will be a relatively new characterization of so-called survival functions of probability measures on (subsets of) Rn, in this case on Nn0

Higher order monotonic functions of several variables

Multivariate functions with a specific degree of higher order monotonicity in each variable are introduced. When normalized, they turn out to form a simplex whose extreme points are precisely the tensor products of their univariate counterparts. Under natural conditions on the degrees these functions will be shown to operate on each other

Higher order alternating sequences

Sequences which are alternating of a (finite) higher order n, appropriately normalized, are shown to form a Bauer simplex, and its countably many extreme points are identified. For n = 2 we are dealing with increasing concave sequences. The proof makes use of multivariate co-survival functions of (not necessarily finite) Radon measures

Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences

This paper deals with a surprising connection between exchangeable distributions on {0, 1}n and the recently introduced Lévy-frailty copulas, the link being provided by a new class of multivariate distribution functions called linearly order symmetric. The characterisation theorem for Lévy-frailty copulas is given a new and short (non-combinatorial) proof, and a related result is shown for exchangeable Marshall–Olkin distributions. A common thread in all these considerations is higher order monotonic functions on integer intervals of the form {0, 1, . . . , n}

Functions operating on several multivariate distribution functions

Functions f on [0, 1]^m such that every composition f ◦ (g1, . . . , gm) with d-dimensional distribution functions g1, . . . , gm is again a distribution function, turn out to be characterized by a very natural monotonicity condition, which for d = 2 means ultramodularity. For m = 1 (and d = 2) this is equivalent with increasing convexity

On the compounding of higher order monotonic pseudo-Boolean functions

Compounding submodular monotone (i.e. 2-alternating) set functions on a finite set preserves this property, as shown in 2010. A natural generalization to k-alternating functions was presented in 2018, however hardly readable because of page long formulas. We give an easier proof of a more general result, exploiting known properties of higher order monotonic functions